Magnetic Exchange Interactions in Quantum Dots Containing Electrons and Magnetic Ions

We present a theory of magnetic exchange interactions in quantum dots containing electrons and magnetic ions. We find the interaction between the electron and Mn ion to depend strongly on the number of electrons. It can be switched off for closed shell configurations and maximized for partially filled shells. However, unlike the total electron spin $S$ which is maximized for half-filled shells, we predict the exchange interaction to be independent of the filling of the electronic shell. We show how this unusual effect manifests itself in quantum dot addition and excitation spectrum.

Figure 1

Exchange matrix elements as a function of Mn position $R$ for a QD with ${\omega }_0=4\mathrm{Ry}$, $d=20\AA$ for $s$ shell (a), $p$ shell (b) and (c), and $d$ shell (d), (e), and (f). Empty circle indicates the position of Mn ion used in our calculation.

Figure 2

Local Zeeman spin splitting $\Delta =|(E_c-E_e)|$ vs number of electrons $N$ for a QD with ${\omega }_0=4\mathrm{Ry}$, $d=20\AA$, and $R=l_0$, calculated using a limited Hilbert space. The corresponding exchange parameters are indicated by $J_{ss}$, $J_{pp}$, and $J_{dd}$, respectively. Inset shows results of numerical calculation including all configurations of $s$, $p$, and $d$ shells.

Figure 3

(a) Difference of the chemical potential $\Delta \mu ={\mu }_c-{\mu }_e$ between a single Mn ion doped and undoped quantum dot as a function of the number of electrons $N$. Spectral function for adding an electron with spin-down to $s$ level of an empty QD (b), and to $p$ shell of $N=2$ QD (c) with ${\omega }_0=4\mathrm{Ry}$ and $d=20\AA$. For comparison, the spectral function of QD without Mn ion is shown as a dotted line in (b) and (c).